Method of text similarity measurement

ABSTRACT

In one aspect, the present invention provides a for estimating the similarity between at least two portions of text including the steps of forming a set of syntactic tuples, each tuple including at least two terms and a relation betweeen the two terms; classifying the relation between the terms in the tuples according to a predefined set of relations; establishing the relative agreement between syntactic tuples from the portions of text under comparison according to predefined classes of agreement; calculating a value representative of the similarity between the portions of text of each of the classes of agreement; and establishing a value for the similarity between the portions of text by calculating a weighted sum of the values representative of the similarity between the portions of text for each of the classes of agreement. Preferaly, the step of calculating a value representative of the similarity between the portions of text for each of the classes of agreement includes a weighting based upon the number of matched terms occurring in particular parts of speech in which the text occurs. It is also preferred that the step of calculating a value representative of the similarity between the portions of text for each of the classes of agreement include the application of a weighting factor to the estimate of similarity for each of the classes of agreement and the parts of speech in which matched terms occur.

FIELD OF THE INVENTION

The present invention relates to an improved method for the measurementof text similarity based on syntactic relation vectors referred to inthe specification as 3-tuples. The method may be used to measure thesimilarity or degree of overlap between two portions of text.

BACKGROUND OF THE INVENTION

Text similarity measurement generally refers to methods of determiningthe degree of overlap between two portions of text. These methods areusually implemented during a search for a document to judge how well adocument matches a given search query. However, these methods may alsobe used in other information retrieval problems such as documentclustering (to match two documents and decide whether they should be inthe same cluster) and the classification of documents.

A method for determining the similarity of portions of text is thecosine measure. For this measure, the portions of text are assumed to berepresented as vectors with terms as the coordinates after appropriateweighting. An estimate of similarity is determined as the cosine of theangle between the vectors. Often the cosine is calculated as the dotproduct of the vectors after normalization.

The cosine measure is generally considered to have advantageousproperties such as fast computation and a symmetric and simple range ofvalues [0,1]. However, with this method it is assumed that the terms areindependent and each portion of text is simply treated as a ‘bag ofterms’.

As a result, the method is limited in its ability to accurately capturethe degree of similarity between two portions of text.

Other methods for determining the similarity between two portions oftext are available, however, they generally have the sameabove-mentioned disadvantage as the cosine measure.

Another type of method for determining the similarity between twoportions of text uses the co-occurrence of term pairs in each of theportions. For this type of method, a co-occurrence thesaurus is firstconstructed from the text portions. This method captures all theco-occurring term pairs in the portions. In this type of method thesimilarity is determined by reference to the amount of overlap betweenco-occurring term pairs. Co-occurrence based measures capture some ofthe term dependencies, but this method can be used only if there is asufficiently large portion of text available for the generation of aco-occurrence thesaurus or if the co-occurrence thesaurus is alreadyavailable.

U.S. Pat. No. 5,297,039 (Kanaegami et.al) proposes a similarity measureusing syntactic relations between the terms. Text is parsed first toextract an ‘analysis to network’ that consists of triplets of the form‘(relation, element 1, element 2)’. The elements correspond to the nounsand the relation is a term (usually a verb) syntactically close to theelements 1 and 2. The similarity is measured by a sum of termagreements, pair agreements and line agreements between thecorresponding analysis networks, after suitable weighting. Since therelations are themselves terms extracted from the text, this method doesnot overcome the problem of synonymity. Accordingly, it is difficult tocalculate the term, pair and line agreements accurately.

The above discussion of documents, acts, materials, devices, articles orthe like is included in this specification solely for the purpose ofproviding a context for the present invention. It is not suggested orrepresented that any or all of these matters formed part of the priorart base or were common general knowledge in the field relevant to thepresent invention as it existed before the priority date of each claimof this application.

SUMMARY OF THE INVENTION

In one aspect, the present invention provides a method for estimatingthe similarity between at least two portions of text including the stepsof:

-   -   forming a set of syntactic tuples, each tuple including at least        two terms and a relation between the two terms;    -   classifying the relation between the terms in the tuples        according to a predefined set of relations;    -   establishing the relative agreement between syntactic tuples        from the portions of text under comparison according to        predefined classes of agreement;    -   calculating a value representative of the similarity between the        portions of text for each of the classes of agreement; and    -   establishing a value for the similarity between the portions of        text by calculating a weighted sum of the values representative        of the similarity between the portions of text for each of the        classes of agreement.

Preferably, the step of calculating a value representative of thesimilarity between the portions of text for each of the classes ofagreement includes a weighting based upon the number of matched termsoccurring in particular parts of speech in which the text occurs.

It is also preferred that the step of calculating a value representativeof the similarity between the portions of text for each of the classesof agreement include the application of a weighting factor to theestimate of similarity for each of the classes of agreement and theparts of speech in which matched terms occur.

The syntactic tuples may include two terms and a relation between theterms, the relation being classified according to one of the followingclasses:

-   -   relations between a verb and its subject;    -   relations between a verb and its object;    -   relations between a modifier and its modified word; or    -   a circumstantial relationship.

In a particularly preferred embodiment, the method includes tuples ofthe form (relation, term1, term2).

The improved method includes a number of advantages in that the methoddoes not rely upon an assumption that the terms in the portions of textare independent. In addition, the method is not constrained by thenumber of relations and uses generic relations between the terms todetermine the similarity between the portions of text with greateraccuracy. Also, the improved method is not biased by the length of theportions of text that are to be compared.

BRIEF DESCRIPTION OF THE DRAWING

The invention will now be described in greater detail by reference tothe attached drawing and to an Example. It is to be understood that theparticularity of the drawing and the Example does not supersede thegenerality of the foregoing description of the invention.

FIG. 1 is a flow chart showing the steps in the method of the invention.

DETAILED DESCRIPTION

The present invention will now be described in relation to the drawingand an Example. FIG. 1 shows the method of the invention in flowchartform. The method for estimating the similarity between at least twoportions of text begins with the step of extracting a set of syntactictuples, each tuple including at least two terms and a relation betweenthe two terms. The extracted syntactic tuples are compared andclassified based on relative agreements. A value representative of thesimilarity between the portions of text for each of the classes ofagreement is calculated; and a value for the similarity between theportions of text is established by calculating a weighted sum of thevalues representative of the similarity between the portions of text foreach of the classes of agreement.

EXAMPLE

In the Example, a method for determining the similarity of two portionsof text is based upon the formation of syntactic tuples of the form“relation-term1l-term2”, referred to as 3-tuples, where “term1” and“term2” are terms extracted from the portions of text to be compared and“relation” is a relationship between the terms.

Preferably, the “relation” is one of a pre-defined set of relations.

3-tuples for each of the portions of text to be compared may beextracted through a three step process including the following steps:

Text Pre-Processing

In this step, redundant parts in the portions of text, such as tabulardata in appendices, are identified and removed.

Morphological Analysis

This particular step can be subdivided into a word analysis stepfollowed by a lexical pattern matching step. Preferably, the wordanalysis step analyses each word in the portions of text using a set ofdictionaries and a morphology grammar. Generally, each word in adictionary is associated with a set of linguistic features.

The morphology grammar defines how phenomena such as punctuations andinflections are to be interpreted by the word analyser. The output ofthe word analysis step is a list of nodes each corresponding to a wordin the portion of text under analysis and containing its linguisticfeatures.

The lexical pattern matching step uses a set of dictionaries with eachentry describing a particular word sequence pattern that is to bematched with the list of nodes output by the word analysis step. Thisstep is used to locate domain terms, idioms and phrases in the portionof text.

Structural Analysis

In the Example, the patterns output from the morphological analysis stepare considered as being related according to a set of predefinedrelationships as set out below:

-   -   ARG0: relations between a verb and its subject, or a subject in        the parser tree of one sentence.    -   ARG1: relations between a verb and its object, or an object in        the parser tree of one sentence.    -   ATTR: relations between a modifier and its modified word.    -   CICR: circumstantial relationship (relations other than the        above).

The structural analysis step may proceed by initially representing thepatterns as nodes and then constructing a directed acyclic graph ofthese nodes. The edges of this directed acyclic graph are extracted andbecome the 3-tuples for the purpose of this Example.

The relations in the 3-tuples are generic and are determined based upona defined set of relations. Accordingly, even though the same “textmeaning” may be expressed in different forms, they will be normalizedinto the same set of 3-tuples with this method.

Different Types of 3-Tuple Agreements

In the Example, three separate types of “agreement” between 3-tuples aredefined. In this embodiment, they are referred to as Class A (fullagreement), Class B (partial agreement) and Class C (term agreement).The following definitions are established for purposes of illustration:

Two portions of text, say D_(i) and D_(j), are represented by a vectorof 3-tuples, that is:D_(i)={T_(i1), T_(i2), . . . , T_(in)}, andD_(j)={T_(j1), T_(j2), . . . , T_(jn)},whereT_(i1)={T_(i1).relation, T_(i1).term1, T_(i1). Term2} andT_(j1)={T_(j1).relation, T_(j1).term1, T_(j1). Term2} are 3-tuples.

The classes of agreement between 3-tuples are defined in the Example asfollows:

Class A (Full Agreement)

In the Example, two 3-tuples are said to belong to “class A” if and onlyif they are identical.

-   -   i.e. T_(i) and T_(j) belong to class A        -   if and only if            (T _(i).relation==T _(j).relation) AND            (T _(i).term1==T _(j).term1) AND            (T _(i).term2==T _(j).term2)            Class B (Partial Agreement)

In the Example, two 3-tuples are said to belong to “class B” if and onlyif exactly two of corresponding elements in the 3-tuples are identical.

-   -   i.e. T_(i) and T_(j) belong to class B        if and only if        ((T _(i).relation==T _(j).relation) AND (T _(i).term1==T        _(j).term1) AND (T _(i).term2!=T _(j).term2)) OR        ((T _(i).relation==T _(j).relation) AND (T _(i).term1!=T        _(j).term1) AND (T _(i).term2==T _(J).term2)) OR        ((T _(i).relation!=T _(j).relation) AND (T _(i).term1==T        _(j).term1) AND (T _(i).term2==T _(J).term2)) OR        Class C (Term Agreement)

In the Example, two 3-tuples are said to belong to “class C” if and onlyif exactly one of the corresponding terms in the 3-tuples are identical.

-   -   i.e. T_(i) and T_(j) belong to class C        if and only if        ((T _(i).relation!=T _(j).relation) AND (T _(i).term1==T        _(j).term1) AND (T _(i).term2!=T _(j).term2)) OR        ((T _(i).relation!=T _(j).relation) AND (T _(i).term1!=T        _(j).term1) AND (T _(i).term2==T _(J).term2))

Preferably, when terms are matched, they are differentiated based on thePart of Speech (POS) in which the terms occur. In the Example, there arethree definitions for POS which are ultimately used in the overallmeasure of similarity between portions of text.

POS1 (Part of Speech 1)

In the Example, two 3-tuples are categorised as POS1 if the POS of eachmatching term is a “NOUN”, that is:(T _(i).term1==T _(j).term1) AND(T _(i).term1.POS==NOUN AND T _(j).term1.POS==NOUN)AND(T _(i).term2==T _(j).term2) AND(T _(i).term2.POS==NOUN AND T _(j).term2.POS==NOUN)POS2 (Part of Speech 2)

In the Example, two 3-tuples are categorised as POS2 if the POS ofexactly one matching term is a “NOUN”, that is:((T _(i).term1==T _(j).term1) AND(T _(i).term1.POS==NOUN AND T _(j).term1.POS==NOUN)AND(T _(i).term2==T _(j).term2) AND(T _(i).term2.POS!=NOUN OR T _(j).term2.POS!=NOUN))OR((T _(i).term1==T _(j).term1) AND(T _(i).term1.POS!=NOUN OR T _(j).term1.POS!=NOUN)AND(T _(i).term2==T _(j).term2) AND(T _(i).term2.POS==NOUN AND T _(j).term2.POS==NOUN))POS3 (Part of Speech 3)

In the Example, two 3-tuples are categorised as POS3 if they neitherbelong to POS1 nor POS2.

Based upon the definitions set out above, a similarity measure in theExample is established as follows:

Two portions of text that will undergo a similarity comparison have3-tuple vectors formed which are referred to as D_(i) and D_(j), whichmay be represented as:D_(i)={T_(i1), T_(i2), . . . , T_(iNi)}andD_(i)={T_(j1), T_(j2), . . . , T_(jNj)}where N_(i) and N_(j) are the length of vectors D_(i) and

The number of 3-tuples that match according to the Class A definitionmay be represented as nA. Similarly, nB and nC may be used to representthe number of 3-tuples matching according to Class B and Class Cdefinitions respectively. In the Example, if a 3-tuple occurs two ormore times, the count is taken to be the minimum of the number of timesit occurs in D_(i) and D_(j).

If nA_(i), (i=1,2,3) is used to represent the number of 3-tuplessatisfying both Class A and POSi, (i=1,2,3), then nA₁+nA₂+nA₃=nA.Similarly, by using nB_(i) and nC_(i), (i=1,2,3) to represent the numberof 3-tuples satisfying both Class B and POSi and Class C and POSirespectively, then measures (sim_(A), sim_(B) and sim_(C)) that may beused to determine the amount of overlap in terms of nA_(i), nB_(i) andnC_(i), may be stated as:

$\begin{matrix}{{{sim}_{A}\left( {D_{i},D_{j}} \right)} = \frac{\sum\limits_{i = 1}^{3}\;{n\;{A_{i} \cdot w_{i}}}}{\min\left( {N_{i},N_{j}} \right)}} \\{{{sim}_{B}\left( {D_{i},D_{j}} \right)} = \frac{\sum\limits_{i = 1}^{3}\;{n\;{B_{i} \cdot w_{i}}}}{3 \cdot {\min\left( {N_{i},N_{j}} \right)}}} \\{{{sim}_{C}\left( {D_{i},D_{j}} \right)} = \frac{\sum\limits_{i = 1}^{3}\;{n\;{C_{i} \cdot w_{i}}}}{2 \cdot {\min\left( {N_{i},N_{j}} \right)}}}\end{matrix}$where w₁, w₂ and w₃ are weightings corresponding to the POS1, POS2 andPOS3 matchings respectively, such that w₁₊w₂₊w₃₌1.00 and w₁≧w₂≧w₃.

It will be recognised by persons skilled in the art thatsim_(C)(D_(i),D_(j)) provides a measure of the overlap of terms,sim_(B)(D_(i),D_(j)) provides a measure of the overlap of lexicallyclose pairs (relations and terms) and sim_(A)(D_(i),D_(j)) provides ameasure of the overlap of 3-tuples.

It will also be recognised that these measures used in the Example arenormalised (ie they take values in the range of (0,1) with identicalportions of text having a value of 1 and portions having no matched3-tuples having a value of 0. In addition, it should be noted that thesemeasures are symmetric.

In the Example, a weighted sum of these measures is calculated asfollows;sim(D _(i) ,D _(j))=w _(a)×sim_(A)(D _(i) ,D _(j))+w _(b)×sim_(B)(D _(i),D _(j))+w _(c)×sim_(C)(D _(i) ,D _(j))

In the calculation of the weighted sum in the Example, the weights arechosen such that w_(a)+w_(b)+w_(c)=1.0 and w_(a)>w_(b)≧w_(c) (for eg,w_(a)=½, w_(b)=⅓, and w_(c)=⅙. Hence this gives rise to a class ofmeasures for various choices of weights.

As the three measures sim_(A), sim_(B) and sim_(C) compute the overlaprespectively with decreasing accuracy, the choice of decreasingweighting factors in the Example is warranted.

The weighted sum provides a measure of the similarity between twoportions of text that can compute text similarity accurately and withoutany requirement for an external dictionary or thesaurus.

Finally, It will be appreciated that there may be other modificationsand alterations made to the configurations described herein that arealso within the scope of the present invention.

1. A method for estimating the similarity between at least two portionsof text, said method comprising the steps of: receiving said at leasttwo portions of text; forming a set of syntactic tuples from saidportions of text, each tuple comprising two terms and a relation betweenthe two terms; classifying the relation between the terms in the tuplesaccording to a predefined set of relations; predefining classes ofagreement between tuples under comparison, comprising a class of fullagreement wherein tuples under comparison are identical, a class ofpartial agreement wherein only two of corresponding elements in tuplesunder comparison are identical, and a class of term agreement whereinonly one of corresponding terms in tuples under comparison areidentical; determining a respective class of relative agreement betweeneach pair of syntactic tuples from the portions of text under comparisonaccording to the predefined classes of agreement; calculating a valuerepresentative of the similarity between the portions of text for eachof the classes of agreement, based on the plurality of tuples determinedto belong to the respective class of agreement; and determining andoutputting a measure of the similarity between the portions of text bycalculating a weighted sum of the values representative of thesimilarity between the portions of text for each of the classes ofagreement.
 2. A method according to claim 1 wherein the step ofcalculating a value representative of the similarity between theportions of text for each of the classes of agreement comprises aweighting based upon the number of matched terms occurring in particularparts of speech.
 3. A method according to claim 2 wherein the step ofcalculating a value representative of the similarity between theportions of text for each of the classes of agreement comprises theapplication of a weighting factor to the estimate of similarity for eachof the classes of agreement and the parts of speech in which matchedterms occur.
 4. A method according to claim 1 wherein the syntactictuples comprise two terms and a relation between the terms, the relationbeing classified according to one of the following classes: relationsbetween a verb and its subject; relations between a verb and its object;relations between a modifier and its modified word; or a circumstantialrelationship.
 5. A method according to claim 1 wherein the syntactictuples are of the form (relation, term1, term2).
 6. A method accordingto claim 5 wherein a syntactic tuple of a first portion of text, T_(i),and a syntactic tuple of a second portion of text, T_(j), are associatedwith the class of full agreement of syntactic tuples, class A, if andonly if:(Ti.relation==Tj.relation) AND(Ti.term1==Tj.term1) AND(Ti.term2==Tj.term2).
 7. A method according to claim 6 wherein asyntactic tuple of a first portion of text, T_(i), and a syntactic tupleof a second portion of text, T_(j), are associated with the class ofpartial agreement of syntactic tuples, class B, if and only if:((T _(i).relation==T _(j).relation) AND (T _(i).term1==T _(j).term1) AND(T _(i).term2!=T _(j).term2)) OR((T _(i).relation==T _(j).relation) AND (T _(i).term1!=T _(j).term1) AND(T _(i).term2==T _(J).term2)) OR((T _(i).relation!=T _(j).relation) AND (T _(i).term1==T _(j).term1) AND(T _(i).term2==T _(J).term2)).
 8. A method according to claim 7 whereina syntactic tuple of a first portion of text, T_(i), and a syntactictuple of a second portion of text, T_(j), are associated with the classof term agreement of syntactic tuples, class C, if and only if:((T _(i).relation!=T _(j).relation) AND (T _(i).term1==T _(j).term1))AND (T _(i).term2!=T _(j).term2) OR((T _(i).relation!=T _(j).relation) AND (T _(i).term1!=T _(j).term1))AND (T _(i).term2==T _(j).term2).
 9. A method according to claim 8wherein a syntactic tuple of a first portion of text, T_(i), and asyntactic tuple of a second portion of text, T_(j), are associated witha class of a part of speech, POS1, if the part of speech of eachmatching term in the syntactic tuples is a noun, namely, if:(T _(i).term1==T _(j).term1)AND(T _(i).term1.POS==NOUN AND T _(j).term1.POS==NOUN)AND(T _(i).term2==T _(j).term2) AND(T _(i).term2.POS==NOUN AND T _(j).term2.POS==NOUN).
 10. A methodaccording to claim 9 wherein a syntactic tuple of a first portion oftext, T_(i), and a syntactic tuple of a second portion of text, T_(j),are associated with a class of a part of speech, POS2, if the part ofspeech of one, and only one, matching term in the syntactic tuples is anoun, namely, if:((T _(i).term1==T _(j).term1) AND(T _(i).term1.POS==NOUN AND T _(j).term1.POS==NOUN)AND(T _(i).term2==T _(j).term2) AND(T _(i).term2.POS!=NOUN OR T _(j).term2.POS!=NOUN))OR((T _(i).term1==T _(j).term1) AND(T _(i).term1.POS!=NOUN OR T _(j).term1.POS!=NOUN)AND(T _(i).term1==T _(j).term1) AND(T _(i).term1.POS==NOUN AND T _(j).term1.POS==NOUN).
 11. A methodaccording to claim 10 wherein a syntactic tuple of a first portion oftext, T_(i), and a syntactic tuple of a second portion of text, T_(j),are associated with a class of a part of speech, POS3, if they are notassociated with either class POS1 or POS2.
 12. A method according toclaim 11 wherein the step of calculating a value representative of thesimilarity between the portions of text for each of the classes ofagreement comprises a weighting based upon the number of matched termsoccurring in particular parts of speech, the calculation of the valuefor Class A comprising parameters relating to the number of matchingsyntactic tuples according to Class A and POS1, nA₁, the number ofmatching syntactic tuples according to Class A and POS2, nA₂, and thenumber of matching syntactic tuples according to Class A and POS3, nA₃,and separate weighting factors w₁, w₂ and w₃ also being applied to thePOS1, POS2 and POS3 matchings, the value, sim_(A)(D_(i),D_(J)), beingcalculated as:${{sim}_{A}\left( {D_{i},D_{j}} \right)} = \frac{\sum\limits_{i = 1}^{3}\;{n\;{A_{i} \cdot w_{i}}}}{\min\left( {N_{i},N_{j}} \right)}$where min(N_(i),N_(j)) represents the lesser value of the two lengths ofthe vectors of syntactic tuples formed for the portions of text D_(i)and D_(j), respectively.
 13. A method according to claim 12 wherein thestep of calculating a value representative of the similarity between theportions of text for each of the classes of agreement comprises aweighting based upon the number of matched terms occurring in particularparts of speech, the calculation of the value for Class B comprisingparameters relating to the number of matching syntactic tuples accordingto Class B and POS1, nB₁, the number of matching syntactic tuplesaccording to Class B and POS2, nB₂, and the number of matching syntactictuples according to Class B and POS3, nB₃, and separate weightingfactors w₁, w₂ and w₃ also being applied to the POS1, POS2 and POS3matchings, the value, sim_(B)(D_(i),D_(j)), being calculated as:${{sim}_{B}\left( {D_{i},D_{j}} \right)} = \frac{\sum\limits_{i = 1}^{3}\;{n\;{B_{i} \cdot w_{i}}}}{3 \cdot {\min\left( {N_{i},N_{j}} \right)}}$where min(N_(i),N_(j)) represents the lesser value of the two lengths ofthe vectors of syntactic tuples formed for the portions of text D_(i)and D_(j) respectively.
 14. A method according to claim 13 wherein thestep of calculating a value representative of the similarity between theportions of text for each of the classes of agreement comprises aweighting based upon the number of matched terms occurring in particularparts of speech, the calculation of the value for Class C comprisingparameters relating to the number of matching syntactic tuples accordingto Class C and POS1, nC₁, the number of matching syntactic tuplesaccording to class C and POS2, nC₂, and the number of matching syntactictuples according to Class C and POS3, nC₃, and separate weightingfactors w₁, w₂ and w₃ also being applied to the POS1, POS2 and POS3matchings, the value, sim_(c)(D_(i),D_(j)), being calculated as:${{sim}_{C}\left( {D_{i},D_{j}} \right)} = \frac{\sum\limits_{i = 1}^{3}\;{n\;{C_{i} \cdot w_{i}}}}{2 \cdot {\min\left( {N_{i},N_{j}} \right)}}$where min(N_(i), N_(j)) represents the lesser value of the two lengthsof the vectors of syntactic tuples formed for the portions of text D_(i)and D_(j) respectively.
 15. A method according to claim 12 wherein theweighting factors, w₁, w₂ and w₃ are such that:w ₁ +w ₂ +w ₃=1.00 and w ₁ ≧w ₂ ≧w ₃.
 16. A method according to claim 13wherein the weighting factors, w₁, w₂ and w₃ are such that:w ₁ +w ₂ +w ₃=1.00 and w ₁ ≧w ₂ ≧w ₃.
 17. A method according to claim 14wherein the weighting factors, w₁, w₂ and w₃ are such that:w ₁ +w ₂ +w ₃=1.00 and w ₁ ≧w ₂ ≧w ₃.
 18. A method according to claim 17wherein the step of determining and outputting a measure of thesimilarity between the portions of text by calculating a weighted sum ofthe values representative of the similarity between the portions of textfor each of the classes of agreement, sim(D_(i),D_(j)), is calculatedas:sim(D _(i) ,D _(j))=w _(a)×sim_(A)(D _(i) ,D _(j))+w _(b)×sim_(B)(D _(i),D _(j))+w _(c)×sim_(C)(D _(i) ,D _(j)) where w_(a), w_(b) and w_(c) areweighting factors.
 19. A method according to claim 18 wherein theweighting factors, wa, wb and wc are such that:w _(a) +w _(b) +w _(c)=1.0 and w _(a) ≧w _(b) ≧w _(c).